Tank Study

Tank

Both the adult and neonatal head tanks created in chapter 2 were used in this study. The methodology used first in section 2.2 and expended upon in section 3.2.4 was once again employed in these experiments with 0.2 % saline and skull present in all cases.

Perturbations

The three perturbation locations,Posterior, Lateraland Centraldescribed in section 3.2.4, as proposed by Malone, Jehl, Arridge,et al. [59], were again used during the adult tank recordings. Two locations were added in the neonatal study, as spatial variations arising from the neonatal skull are less understood in the literature. To investigate the effects of the frontal or metopic suture between the two frontal bones, a perturbation was added in theAnteriorposition. TheCentralperturbation was shifted upwards by 20 mm towards the corona capitis, and a newCaudalposition was added in the centre of the head level with the temporal fontanelles. This new perturbation is located at a point the furthest possible distance away from the electrodes, and thus represents the “worst case” scenario for reconstructing images.

(a)Adult Tank (b)Neonatal Tank

Figure 4.3: Perturbation locations used in tank study,(a)adult tank based on positions from Malone,

Jehl, Arridge,et al.[59]and(b)neonatal tank with two additional locations

The perturbations used in this study were spheres of polyurethane sponge (5 % density), chosen to match those used by Tidswell, Gibson, Bayford,et al.[74]to provide an impedance change of≈12 %, similar to those found during epileptic seizures by Rao, Gibson, and Holder

[28]and Elazar, Kado, and Adey[84].

For the adult tank the perturbations were 25 mm in diameter, whereas for the neonatal experiments the volume of the perturbation was scaled based on the relative volume of the skull cavity, giving a perturbation of 19 mm diameter. The air was removed from within the

sponge through repeated compression in saline, and was kept submerged in saline between recordings. A baseline was recorded between each perturbation.

EIT System Settings

The ScouseTom system, described in section 3.1.1, was used for all recordings as it was found to have the best performance in the tank study in chapter 3. The “optimal” and “offset” injection protocols described in section 2.2.7 were used in this study for the adult and neonatal tanks respectively. The current injection was increased to 250µA which, due to IEC 60601 regulations, necessitated an increase of carrier frequency to 1.76 kHz. The injection time of 75 ms was maintained, resulting in a scan time of≈2 s for both the “optimal” and

“offset” protocols. The protocol was repeated 20 times for each background and perturbation recording, the same number used by Fabrizi, McEwan, Oh,et al.[58].

Image Reconstruction

Images were generated using two separate reconstruction methods in this study. First the implementation of first order Tikhonov regularisation by Markus Jehl[63], described in sec- tion 3.2.4 was used. Secondly a zeroth order Tikhonov regularisation noise-based correction developed by Aristovich, Santos, Packham,et al.[61]was used to reconstruct images. Unlike first order Tikhonov regularisation, zeroth order does not bias the solution towards finding small connected perturbations, and thus does not include prior information regarding the ex- pected changes. Zeroth-order Tikhonov minimises the L2 norm magnitude of the conductivity changes:

ˆ

δσ=argmin_δσkδv−Jδσk2+λkδσk2 (4.1)

where ˆδσis the estimated conductivity change,δvis the boundary voltage change andλ is a hyper-parameter set using cross validation.

The optimal value forλwas the one which minimised the cross validated error, and was found using leave-one-out cross validation using a range of 3000 logarithmically spacedλs between×10−20and×10−2[61]. Once the optimalλwas found, the Jacobian matrix was

inverted using:

J−_λ1= JTJ+λI−1JT (4.2) These images were post processed using a noise based correction approach developed by Aristovich, Santos, Packham,et al.[61]. The standard deviation of the conductivity change in each hexahedron in the inverted JacobianJ−_λ1was calculated for random Gaussian Noise in the voltage measurements:

δσn=J−_λ1δvn (4.3) whereδv_n is normal distribution of mean 0 and standard deviation equal to that of the

noise in the measured voltages. δσnis the conductivity change matrix of sizem(number of elements) byk(number of samples) caused by the noise in the voltage changeδv_n on the

electrodes. For the actual conductivity change in each element at-score can be computed:

t_i= δσ

i

〈δσi

n−δσ¯ni〉

(4.4)

The values in the reconstructions with this algorithm therefore represent the noise based- scaled values, which are unitless and do not reflect the absolute conductivity in S/m.

The≈4 million element “fine” meshes, described in section 2.2.5 were used to compute

the forward solution and generate the Jacobian matrices using PEITS[63]. For both the adult and neonatal head meshes, the Jacobian matrixJ_σcomputed on the fine mesh was summed into geometrically regular cubesJ_{he x}, as was performed in section 2.2.5.

Separate hexahedral meshes were generated for use with zeroth and first order Tikhonov. One of the benefits of the simpler regularisation of zeroth-order Tikhonov is that it does not require the use of the memory-intensive gSVD (section 3.2.4), which limits the number of elements in the hexahedral mesh. Thus the number of elements in the meshes used for zeroth order reconstructions are significantly greater than the corresponding meshed for first order. For the adult head, the 2526 10 mm element mesh created in section 3.2.4, and a 76996 3 mm element mesh were used for first order and zeroth order reconstructions respectively. For the neonatal head a 4012 6 mm element mesh was used for the first order reconstructions, and a 83839 mesh of 1.5 mm elements was used in the zeroth order reconstructions.

The hyper-parameterλwas found using cross validation as previously described for zero order Tikhonov. A constantλ=4.6·10−4 as described in section 3.2.4 was used for the

first-order reconstructions in the adult head tank. As with the adult tank, the L-curve was not pronounced enough to find the value automatically for each neonatal tank reconstruction. Thus the lambda was again fixed, at a value ofλ=2.5·10−4, giving similar solution norm ||x_||2and residual norm_||Jx₋v_||2similar to those in the adult tank of around 0.1 and 1.0_·10−4 respectively.

Image Quantification

Images were quantified using the same algorithms written in section 3.2.4. Thereconstructed perturbationwas first identified before theLocalisation Error,Shape ErrorandImage Noise metrics were calculated with respect to the “ideal” reconstructed perturbations. Additionally, the localisation error was expressed as the norm of the displacement of the centre of mass in

mm, to give a more intuitive understanding of the error.

Boundary Voltage Preprocessing

As this study does not have many commonalities with previous tank experiments within the group, with different geometry, hardware, mesh, forward and inversion, it was unclear the extent to which the conclusions previously drawn regarding boundary voltage preprocessing were relevant. Therefore the processing was kept to a minimum. Occasionally during mea- surements, a change in the DC offset across all channels would occur, which would corrupt all measurements for a single injection. These were removed from the calculation of the average boundary voltage by removing measurements six times above the median for that channel. This was largely unused throughout the experiments, being invoked a total of three times across 16 measurements.